Statistics

Visible Points on Curves over Finite Fields

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For a prime 𝑝 and an absolutely irreducible modulo 𝑝 polynomial 𝑓(U,V)
∈ ℤ[U,V] we obtain an asymptotic formulas for the number of solutions to
the congruence 𝑓(𝑥,𝑦) ≡ a (mod 𝑝) in positive integers 𝑥 ⩽ X, 𝑦 ⩽
Y, with the additional condition 𝗀cd(𝑥,𝑦)=1. Such solutions have a natural
interpretation as solutions which are visible from the origin. These formulas
are derived on average over 𝑎 for a fixed prime 𝑝, and also on average over
𝑝 for a fixed integer 𝑎.